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Physica D 07 (005) Wave collapse in a class of nonlocal nonlinear Schrödinger equations Mark Ablowitz a, İlkay Bakırtaş b, Boaz Ilan a, a Department of Applied Mathematics, University of Colorado,

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Physica D 07 (005) Wave collapse in a class of nonlocal nonlinear Schrödinger equations Mark Ablowitz a, İlkay Bakırtaş b, Boaz Ilan a, a Department of Applied Mathematics, University of Colorado, Campus Box 56, Boulder, CO , USA b Department of Mathematics, Istanbul Technical University, Maslak 34469, Istanbul, Turkey Received 10 October 004; received in revised form 30 March 005; accepted 3 June 005 Available online June 005 Communicated by A.C. Newell Abstract Wave collapse is investigated in nonlocal nonlinear Schrödinger (NLS) systems, where a nonlocal potential is coupled to an underlying mean term. Such systems, here referred to as NLS-Mean (NLSM) systems, are also known as Benney Roskes or Davey Stewartson type and they arise in studies of shallow water waves and nonlinear optics. The role of the ground-state in global-existence theory is elucidated. The ground-state is computed using a fixed-point method. The critical-powers for collapse predicted by the Virial Theorem, global-existence theory, and by direct numerical simulations of the NLSM are found to be in good agreement with each other for a wide range of parameters. The ground-state profile in the water-wave case is found to be generically narrower along the direction of propagation, whereas in the optics case it is generically wider along the axis of linear polarization. In addition, numerical simulations show that NLSM collapse occurs with a quasi self-similar profile that is a modulation of the corresponding astigmatic ground-state, which is in the same spirit as in NLS collapse. It is also found that NLSM collapse can be arrested by small nonlinear saturation. 005 Elsevier B.V. All rights reserved. Keywords: Blowup; Singularity formation; Modulated nonlinear waves 1. Introduction Nonlinear waves problems are of wide physical and mathematical interest and arise in a variety of scientific fields such as nonlinear optics, fluid dynamics, plasma physics, etc. (cf. [7,36]). The solutions of the governing nonlinear Corresponding author. Tel.: ; address: (B. Ilan) /$ see front matter 005 Elsevier B.V. All rights reserved. doi: /j.physd M. Ablowitz et al. / Physica D 07 (005) waves equations often exhibit important phenomena, such as stable localized waves (e.g., solitons), self-similar structures, chaotic dynamics and wave singularities such as shock waves (derivative discontinuities) and/or wave collapse (i.e., blowup) where the solution tends to infinity in finite time or finite propagation distance. A prototypical equation that arises in cubic media, such as Kerr media in optics, is the (+1)D focusing cubic nonlinear Schrödinger equation (NLS), iu z (x, y, z) + 1 u + u u = 0, u(x, y, 0) = u 0 (x, y), (1) where u is the slowly-varying envelope of the wave, z is the direction of propagation, 1 (x, y) are the transverse directions, u u xx + u yy, and u 0 is the initial conditions. Remarkably, in 1965 Kelley [3] carried out direct numerical calculations of (1) that indicated the possibility of wave collapse. In 1970, Vlasov et al. [34] proved that solutions of Eq. (1) satisfy the following Virial Theorem (also called Variance Identity) d dz (x + y ) u = 4H, H = 1 ( u 0 u 0 4 ), () where ( x, y ), the integrations are carried over the (x, y) plane, and H, which is a constant of motion, is the Hamiltonian of Eq. (1). Using the Virial Theorem, Vlasov et al. concluded that the solution of the NLS can become singular in finite distance (or time), because a positive-definite quantity could become negative for initial conditions satisfying H 0. On the other hand, Weinstein [35] showed that when the power (which is also conserved) is sufficiently small, i.e., N = u 0 = constant N c 1.863π, the solution exists globally, i.e., for all z 0. Therefore, a sufficient condition for collapse is H 0while a necessary condition for collapse is N N c. Weinstein also found that the ground-state of the NLS plays an important role in the collapse theory. This ground-state is a stationary solution of the form u = R(r)e iz, such that R is radially-symmetric, positive, and monotonically decaying. Papanicolaou et al. [7] studied the singularity structure near the collapse point and showed asymptotically and numerically that collapse occurs with a (quasi) self-similar profile. The readers are referred to [30] for a comprehensive review of related studies. Recent research by Merle and Raphael [6] further elaborated on the collapse behavior of NLS Eq. (1) and related equations, allowing for detailed understanding of the self-similar asymptotic profile. Furthermore, Gaeta and coworkers [4] recently carried out detailed optical experiments in cubic media that reveal the nature of the singularity formation and showed experimentally that collapse occurs with a self-similar profile. On the other hand, there are considerably fewer studies of wave collapse that arises in nonlinear media, whose governing system of equations have quadratic nonlinearities, such as water waves and χ () nonlinear-optical media. Here we discuss a class of such systems, denoted as NLS-Mean (NLSM) systems, which are sometimes referred to as Benney and Roskes [8] or Davey and Stewartson [13] type. The physical derivation of NLSM systems in water waves and nonlinear optics is reviewed in Section. Broadly speaking, the derivation of NLSM systems is based on an expansion of the slowly-varying (i.e., quasi-monochromatic) wave amplitude in the first and second harmonics of the fundamental frequency, as well as a mean term that corresponds to the zeroth harmonic. This leads to a system of equations that describes the nonlocal-nonlinear coupling between a dynamic field that is associated with the first harmonic (with a cascaded effect from the second harmonic), and a static field that is associated with the mean term (i.e., the zeroth harmonic). For the physical models considered in this study, the general NLSM system can be written in the following non-dimensional form, iu z + 1 (σ 1u xx + u yy ) + σ u u ρuφ x = 0, φ xx + νφ yy = ( u ) x, (3) where u(x, y, t) corresponds to the field associated with the first-harmonic, φ(x, y, t) corresponds to the mean field, σ 1 and σ are ±1, and ν and ρ are real constants that depend on the physical parameters. It is well-known 1 In this study z plays the role of the evolution variable (i.e., like time). 3 M. Ablowitz et al. / Physica D 07 (005) that System (3) can admit collapse of localized waves when σ 1 = σ = 1 and ν 0. In that case, the governing equations are iu z + 1 u + u u ρuφ x = 0, φ xx + νφ yy = ( u ) x, (4a) (4b) where ν 0 and ρ is real, and the initial conditions are u(x, y, 0) = u 0 (x, y), φ(x, y, 0) = φ 0 (x, y), such that Eq. (4b) is satisfied at z = 0, i.e., φ 0,xx + νφ 0,yy = ( u 0 ) x. The goal of this study is to further investigate the collapse dynamics in the NLSM System (4). We note that System (4) reduces to the classical NLS Eq. (1) when ρ = 0, because in that case the mean field φ does not couple to the harmonic field u in Eq. (4a). In addition, when ν = 0 Eq. (4b) gives that φ x = u and, therefore, Eq. (4a) reduces to a classical NLS Eq. (1) with the cubic term (1 ρ) u u. As we shall see, in optics ρ 0, whereas in water waves ρ 0. In either case, i.e., when ρ 0, the NLSM System (4) is a nonlocal system of equations. Indeed, since ν 0, Eq. (4b) can be solved as φ(x, y, z) = G(x x,y y ) x u(x,y,z) dx dy, where G(x, y) is the usual Green s function. For Eq. (4b) G(x, y) = (4π) 1 log(x + y /ν), which corresponds to a strongly-nonlocal function φ. While one might have expected the strong-nonlocality in the NLSM to arrest the collapse process, generally speaking, that is not the case for System (4). Moreover, there is a striking mathematical similarity between collapse dynamics in the NLS and NLSM cases. The paper is organized as follows: (1) In Section NLSM systems in water waves and in nonlinear optics are discussed. () In Section 3 the theory of collapse and global existence in NLS and NLSM equations is reviewed. In addition, the Hamiltonian is used to explain why collapse in the case of water waves (ρ 0) is relatively easier to attain, and also occurs more quickly, than in the case of nonlinear optics (ρ 0). (3) Using global existence theory and numerical calculations of the ground-state, in Section 4 the necessary condition for collapse is explored in terms of the parameters ν and ρ in the NLSM System (4). Using the Virial Theorem and the Hamiltonian, a sufficient condition for collapse is found for Gaussian input beams, explicitly in terms of ν, ρ, and the input power. These theoretical results are found to be consistent with numerical simulations of the NLSM System (4) and are also consistent with the numerical results of Crasovan et al. [1] for nonlinear optics (ρ 0). In addition, the effect of input astigmatism in the initial conditions on the critical power for collapse is studied (Section 4.1]). Furthermore, in Section 4. it is shown that the NLSM can admit collapse even without the cubic term [i.e., without u u in Eq. (4a)]. (4) In Section 5 the astigmatism of the NLSM ground-state is explored in the (ν, ρ) parameter space. It is found that the ground-state is relatively more astigmatic for nonlinear optics (ρ 0) than for water waves (ρ 0). In addition, the dependence of the astigmatism of the ground-state on ν is found to be weaker than its dependence on ρ. (5) In Section 6 simulations of the NLSM System (4) show that the collapsing solution is well described by a quasi self-similar profile that is given in terms of a modulation of the corresponding ground state, a result that is in the same spirit as for the NLS equation and also strengthens the results of Papanicolaou et al. [8]. However, in [8] the ground-state itself was not computed and, in turn, it was not shown numerically that the asymptotic profile approaches the corresponding ground-state. In this study numerical simulations directly show that the collapsing wave approaches a quasi self-similar modulation of the corresponding ground-state. To calculate the ground-state a fixed-point algorithm is used, which has been previously applied in dispersion-managed NLS theory (see Appendix C). M. Ablowitz et al. / Physica D 07 (005) (6) In Section 7 numerical simulations are used to show that NLSM collapse can be arrested by a small saturation of the cubic nonlinearity, a phenomenon that can be explained using the results of Fibich and Papanicolaou [19] for the the perturbed NLS.. NLSM systems in water waves and nonlinear optics Below we review some of the main results from the derivations of NLSM systems, with an emphasis on collapse..1. Water waves In the context of free-surface gravity-capillary water waves, NLSM equations result from a weakly-nonlinear quasi-monochromatic expansion of the velocity potential as φ(x, y, t) ε[ãe i(kx wt) + c.c. + Φ] + ε [Ã e i(kx wt) + c.c.] +, (5) where x is the direction of propagation, y the transverse direction, t the time, ε 1 a measure of the (weak) nonlinearity, Ã, Ã, and Φ are slowly varying functions of (x, y, t), which correspond to the coefficients of the first, second, and zeroth harmonics, respectively, c.c. denotes complex conjugate of the term to its left, and the frequency ω satisfies the dispersion relation ω (κ) = (gκ + Tκ 3 ) tanh(κh), where g is the gravity acceleration, T is the surface tension coefficient, and κ = k + l, where (k, l) are the wave-numbers in the (x, y) directions, respectively. Substituting the wave expansion (5) into the water-wave equations (i.e., Euler s equation with a free surface) and assuming slow modulations of the field in the x and y directions results in a nonlinearly-coupled system for Ã and Φ. After non-dimensionalization, i.e., (A, Φ) = (Ã, Φ)k / gh, one finds the general NLSM system [6] ia τ + λa ξξ + µa ηη = χ A A + χ 1 AΦ ξ, αφ ξξ + Φ ηη = β( A ) ξ, (6a) (6b) where ξ = εk(x c g t), η = εly and τ = ε gk t are dimensionless coordinates, and c g = ω/ k is group velocity. The coefficients λ, µ 0, χ, χ 1 0, α and β 0 are suitable functions of h, T, k, c g, and the second-order dispersion coefficients ω/ k and ω/ l. We note that in the derivation of System (6) Ã is expressed in terms of Ã, which accounts for the fact that A does not appear explicitly in the resulting system. NLSM equations were originally obtained by Benney and Roskes [8] in their study of the instability of wave packets in water of finite depth h, without surface tension. In 1974, Davey and Stewartson [13] studied the evolution of a 3D wave packet in water of finite depth and obtained a different, although equivalent, form of these equations. In 1975 Ablowitz and Haberman [4] studied the integrability of systems such as (6). These integrable systems correspond to the Benney Roskes equations in the shallow water limit. In 1977, Djordevic and Reddekopp [14] extended the results of Benney and Roskes to include surface tension. Subsequently, Ablowitz and Segur [6] investigated System (6) or, equivalently, System (3). They showed that the shallow water limit, i.e., h 0, corresponds to σ 1 ν =±1, and ρ in System (3). The resulting equations agreed with those obtained by Ablowitz and Haberman [4]. Hence, the shallow-water limit of System (6) is integrable and can be obtained from an associated compatible linear scattering system. In [1] these reduced equations were linearized by the inverse scattering transform (see also [3]). Subsequently, Ablowitz and Segur [6] studied the NLSM System (6) in the non-integrable case. In this parameter regime, System (6) can be transformed by a rescaling of variables to System (3) with σ 1 = σ = 1 and ν 0, i.e., A similar observation holds in the optics case mentioned below. 34 M. Ablowitz et al. / Physica D 07 (005) the so called focusing elliptic elliptic case, which, physically speaking, requires sufficiently large surface tension. They found that System (6) preserves the Hamiltonian [ H = λ A ξ + µ A ] 1 [ ( χ) A 4 + αχ 1 η β (Φ ξ) + χ ] 1 β (Φ η), (7) where the first and second integrals correspond to the second-derivative and the nonlinear terms in Eq. (6a), respectively, and the integrations are carried over the (ξ, η) plane. When, in addition to the physical requirements µ 0, β 0, and χ 1 0, one has that λ 0, χ 0, and α 0, the first and second integral terms in (7) are positive and negative-definite, respectively. This corresponds to the self-focusing regime. Clearly, in that case H 0is possible for sufficiently large initial conditions. 3 Furthermore they proved that the following Virial Theorem holds ( ξ ) τ λ + η A = 8H. µ As can be seen, if H 0, the moment of inertia vanishes at a finite time. In that case, as for the NLS case mentioned above, this indicates finite-distance singularity formation. We note that in the same study collapse solutions with the self-similar profile were also investigated, i.e., with A L 1 f ( L x, y L ), where L = L(t) approaches zero during the collapse... Nonlinear optics The electric polarization field of intense laser beams propagating in optical media an be expanded in powers of the electric field as P = χ (1) E + χ () E E + χ (3) E E E +, (8) where E = (E 1,E,E 3 ) the electric field vector and χ (j) are the susceptibility tensor coefficients of the medium. In isotropic Kerr media, where the nonlinear response of the material depends cubically [i.e., through χ (3) and when χ () 0] and instantaneously on the applied field, the dynamics of a quasi-monochromatic optical pulse is governed by the NLS Eq. (1) (cf. [9,3,31]). It turns out that NLSM type equations also arise in nonlinear optics when studying media with a non-zero χ () [even when χ (3) 0], i.e., materials that have a quadratic nonlinear response. Such materials are anisotropic, e.g., crystals whose optical refraction has a preferred direction. Ablowitz et al. [1,] found, from first principles, that NLSM type equations describe the evolution of the electromagnetic field in such quadratically [i.e., χ () ] polarized media. Both scalar and vector (3+1)D NLS systems were obtained. Briefly, in this derivation one assumes a quasi-monochromatic expansion of the x component of the electromagnetic field (which is primarily linearly-polarized), with the fundamental harmonic, second-harmonic, and a mean term as E 1 ε[a e i(kx ωt) + c.c.] + ε [A e i(kx ωt) + c.c. + φ x ] +, (9) where A, A, and φ are slowly varying functions of (x, y, t), which correspond to the first, second, and zeroth harmonics, respectively. Using a polarization field of the form (8) in Maxwell s equations leads to the system of equations [ik Z + (1 α x,1 ) XX + YY kk TT + M x,1 A + M x,0 φ x ]A = 0, (10a) [(1 α x,0 ) XX + YY + s x TT ]φ x α y,0 XY φ y = (N x,1 TT N x, XX ) A, (10b) 3 Note that from Eq. (6b) Φ scales as A, so all the terms in the second integral of (7) scale like A 4. M. Ablowitz et al. / Physica D 07 (005) where α x,0, α x,1, α y,0, and s x depend on the linear polarization term χ (1) ; M x,0, N x,1, and N x, depend on the nonlinear polarization terms χ () and χ (3) ; and M x,1 depends on products of χ () and χ (3). Physically speaking, the dependence of M x,1 on χ () and χ (3) corresponds to the fact that the second-harmonic (i.e., Ã ) is coupled to the first harmonic (i.e., Ã 1 ), a process that is sometimes referred to as optical rectification or cascaded optical effect. However, as in the water-wave case, here too Ã is expressed in terms of Ã, which is why A does not appear explicitly in the resulting system (10). In addition, similar to the water-wave case, the term with M x,0 in System (10a) couples the mean field to the first-harmonic field. Interestingly, when the time dependence in these equations is neglected ( T 0) and for media with a special symmetry class such that α y,0 = 0, it can be seen that, after proper rescaling, the governing system of equation is given by System (4). In[3] these equations were further elucidated and the coefficients described in terms of the electro-optic effect. From the point of view of perturbation analysis, it is interesting to remark that in the expansion of the field in the case of water-waves [i.e., Eq. (5)], the mean term Φ appears as an O(ε) term, whereas in the in the case of optics [i.e., Eq. (9)], the mean term φ x appears as an O(ε ) term. However, the physically measurable quantity in water waves is Φ x, which scales like O(ε ), because Φ is slowly-varying. Therefore, the expansions in the water-wave and optics cases are, in fact, analogous from the viewpoint of perturbation analysis. Wave collapse in such NLSM systems was recently investigated numerically by Crasovan et al. [1]. They solved the following normalized system of equations, iu z + 1 U + U U ρuv = 0, (11a) V xx + νv yy = ( U ) xx, where U is the normalized amplitude of the envelope of the electric field, V the normalized static field, ρ a coupling constant that comes from the combined optical rectification and electro-optic effects, and ν corresponds to the anisotropy coefficient of the medium. They solved System (11) numerically with Gaussian initial conditions for U. The regions of collapse were investigated for various values of the parameters ρ and ν. We note that System (11) is a simple mathematical modification of the NLSM System (4). Indeed, starting with the NLSM System (4), taking the derivative of Eq. (4b) with respect to x, and defining the new variable (potential) V = φ x, one finds that the resulting system is identical to (11). (11b) 3. Global existence, collapse, and the ground-state We begin by briefly outlining some of the known results for the NLS and NLSM equations. Two conserved quantities for the NLS Eq. (1) and NLSM System (4) are the power, i.e., N(u) = u = N(u 0 ), (1) where the integrations (he

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